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Unformatted text preview: MA 36600 LECTURE NOTES: MONDAY, APRIL 13 Basic Theory of Systems of First Order Linear Equations
Solutions to Systems of Linear Diﬀerential Equations. We continue to focus on a system of ﬁrst
order diﬀerential equations in the form
x
1
x
2
x
m =
=
.
.
. p11 (t) x1
p21 (t) x1 = pm1 (t) x1 +
+ p12 (t) x2
p22 (t) x2 + ···
+ ···
.
.
. + pm2 (t) x2 + ··· +
+ p1n (t) xn
p2n (t) xn + pmn (t) xn +
+ g1 (t)
g2 (t) + gm (t) Such a system contains m equations and n variables. When m = n, we can express this as
d
x = P(t) x + g(t).
dt
We focus on a few special cases to gain intuition about the solutions of this system.
Example #1. Consider ﬁrst when m = n = 1. The ﬁrst order equation
x = p(t) x + g (t)
can be solved using integrating factors:
t
µ(t) = exp −
p(τ ) dτ 1
x(t) =
µ(t) =⇒ t µ(τ ) g (τ ) dτ + C . Hence the general solution is in the form x(t) = C x(1) (t) + x(p) (t), where
t
1
d (1)
x(1) (t) =
= exp
p(τ ) dτ
=⇒
x = p(t) x(1)
µ(t)
dt
is a solution to the homogeneous equation, and
(p) x 1
(t) =
µ(t) t µ(τ ) g (τ ) dτ is a particular solution to the nonhomogeneous equation.
Example #2. Consider now when m = n = 2. The equation
x = q (t) x + p(t) x + g (t)
can be transformed into a system of ﬁrst order equations through the substitution
d
x
0
1
0
x=
=⇒
x=
x+
.
x
q (t) p(t)
g (t)
dt Say that we have three functions x1 = x1 (t), x2 = x2 (t), and X = X (t) such that
i. x1 and x2 are solutions to the homogeneous equation
x − p(t) x − q (t) x = 0. ii. Their Wronskian is nonzero, in terms of the function
t
x1 (t) x2 (t)
W x1 , x2 (t) = x1 (t) x (t) − x (t) x2 (t) = det
= C exp
p(τ ) dτ .
2
1
x (t) x (t)
1
2 iii. X is a particular solution to the nonhomogeneous equation X − p(t) X − q (t) X = g (t).
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 Spring '09
 EdrayGoins
 Linear Equations, Equations

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